v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
66 CHAPTER 2. CONVEX GEOMETRYHyperplanes in R mn may, of course, also be represented using matrixvariables.∂H = {Y | 〈A, Y 〉 = b} = {Y | 〈A, Y −Y p 〉 = 0} ⊂ R mn (100)Vector a from Figure 16 is normal to the hyperplane illustrated. Likewise,nonzero vectorized matrix A is normal to hyperplane ∂H ;2.4.2.2 Vertex-description of hyperplaneA ⊥ ∂H in R mn (101)Any hyperplane in R n may be described as the affine hull of a minimal set ofpoints {x l ∈ R n , l = 1... n} arranged columnar in a matrix X ∈ R n×n (65):∂H = aff{x l ∈ R n , l = 1... n} ,dim aff{x l ∀l}=n−1= aff X , dim aff X = n−1= x 1 + R{x l − x 1 , l=2... n} , dim R{x l − x 1 , l=2... n}=n−1= x 1 + R(X − x 1 1 T ) , dim R(X − x 1 1 T ) = n−1whereR(A) = {Ax | ∀x} (120)(102)2.4.2.3 Affine independence, minimal setFor any particular affine set, a minimal set of points constituting itsvertex-description is an affinely independent descriptive set and vice versa.Arbitrary given points {x i ∈ R n , i=1... N} are affinely independent(a.i.) if and only if, over all ζ ∈ R N ζ T 1=1, ζ k = 0 (confer2.1.2)x i ζ i + · · · + x j ζ j − x k = 0, i≠ · · · ≠j ≠k = 1... N (103)has no solution ζ ; in words, iff no point from the given set can be expressedas an affine combination of those remaining. We deducel.i. ⇒ a.i. (104)Consequently, {x i , i=1... N} is an affinely independent set if and only if{x i −x 1 , i=2... N} is a linearly independent (l.i.) set. [ [152,3] ] (Figure 18)XThis is equivalent to the property that the columns of1 T (for X ∈ R n×Nas in (65)) form a linearly independent set. [147,A.1.3]
2.4. HALFSPACE, HYPERPLANE 67A 1A 2A 30Figure 18: Any one particular point of three points illustrated does not belongto affine hull A i (i∈1, 2, 3, each drawn truncated) of points remaining.Three corresponding vectors in R 2 are, therefore, affinely independent (butneither linearly or conically independent).2.4.2.4 Preservation of affine independenceIndependence in the linear (2.1.2.1), affine, and conic (2.10.1) senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (65)holds an affinely independent set in its columns. Consider a transformationT(X) : R n×N → R n×N ∆ = XY (105)where the given matrix Y ∆ = [y 1 y 2 · · · y N ]∈ R N×N is represented by linearoperator T . Affine independence of {Xy i ∈ R n , i=1... N} demands (bydefinition (103)) there exists no solution ζ ∈ R N ζ T 1=1, ζ k = 0, toXy i ζ i + · · · + Xy j ζ j − Xy k = 0, i≠ · · · ≠j ≠k = 1... N (106)By factoring X , we see that is ensured by affine independence of {y i ∈ R N }and by R(Y )∩ N(X) = 0 whereN(A) = {x | Ax=0} (121)
- Page 15 and 16: LIST OF FIGURES 1559 Quadratic func
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- Page 19 and 20: Chapter 1OverviewConvex Optimizatio
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- Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
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- Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
- Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
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- Page 87 and 88: 2.7. CONES 87Thus the simplest and
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2.4. HALFSPACE, HYPERPLANE 67A 1A 2A 30Figure 18: Any one particular point of three points illustrated does not belongto affine hull A i (i∈1, 2, 3, each drawn truncated) of points remaining.Three corresponding vectors in R 2 are, therefore, affinely independent (butneither linearly or conically independent).2.4.2.4 Preservation of affine independenceIndependence in the linear (2.1.2.1), affine, and conic (2.10.1) senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (65)holds an affinely independent set in its columns. Consider a transformationT(X) : R n×N → R n×N ∆ = XY (105)where the given matrix Y ∆ = [y 1 y 2 · · · y N ]∈ R N×N is represented by linearoperator T . Affine independence of {Xy i ∈ R n , i=1... N} demands (bydefinition (103)) there exists no solution ζ ∈ R N ζ T 1=1, ζ k = 0, toXy i ζ i + · · · + Xy j ζ j − Xy k = 0, i≠ · · · ≠j ≠k = 1... N (106)By factoring X , we see that is ensured by affine independence of {y i ∈ R N }and by R(Y )∩ N(X) = 0 whereN(A) = {x | Ax=0} (121)
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